author: Bojanczyk, A. W.; Lutoborski, A.:
shorttitle: Procrustes problem for orthogonal Stiefel matrices
source: SIAM J. Sci. Comput. 21, No. 4, 1291 - 1304 (2000)
rsclass: 65K10; 51F99
keywords: minimizing distance between a point and an ellipsoid, Stiefel manifolds, projections, relaxation methods, Procrustes problem
revtext: Two m-dimensional vectors B and B' define an ellipse E(t) = B cos t + B' sin t. Its distance from a point A is given by the vector A-E(t') of minimal length. The Procrustes problem is to find the ``angle" t'. One can generalize of course, taking p < m different vectors B and replacing the two-dimensional vector (cos t, sin t) of unit length by k < p columns of a p x p orthogonal matrix (the latter k-plets form a set OSt(p,k) of the so called orthogonal Stiefel matrices). The paper proposes a class of relaxation methods for generating sequences of approximations to the k(p-(k+1)/2)-parameteric Stiefel-matrix minimizer. A numerical illustration and geometric interpretation of these methods is offered.